3.836 \(\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=37 \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - ArcCosh[Sqrt[x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0734748, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x],x]

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - ArcCosh[Sqrt[x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.39673, size = 31, normalized size = 0.84 \[ \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1} - \operatorname{acosh}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2)/x**(1/2),x)

[Out]

sqrt(x)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1) - acosh(sqrt(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0293943, size = 49, normalized size = 1.32 \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-2 \sinh ^{-1}\left (\frac{\sqrt{\sqrt{x}-1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x],x]

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - 2*ArcSinh[Sqrt[-1 + Sqrt[x]]/Sqrt
[2]]

_______________________________________________________________________________________

Maple [B]  time = 0.008, size = 72, normalized size = 2. \[ \sqrt{-1+\sqrt{x}} \left ( 1+\sqrt{x} \right ) ^{{\frac{3}{2}}}-\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}-{1\sqrt{ \left ( 1+\sqrt{x} \right ) \left ( -1+\sqrt{x} \right ) }\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ){\frac{1}{\sqrt{-1+\sqrt{x}}}}{\frac{1}{\sqrt{1+\sqrt{x}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(1/2),x)

[Out]

(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(3/2)-(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)-((1+x^
(1/2))*(-1+x^(1/2)))^(1/2)/(1+x^(1/2))^(1/2)/(-1+x^(1/2))^(1/2)*ln(x^(1/2)+(-1+x
)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.37652, size = 35, normalized size = 0.95 \[ \sqrt{x - 1} \sqrt{x} - \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/sqrt(x),x, algorithm="maxima")

[Out]

sqrt(x - 1)*sqrt(x) - log(2*sqrt(x - 1) + 2*sqrt(x))

_______________________________________________________________________________________

Fricas [A]  time = 0.214157, size = 153, normalized size = 4.14 \[ -\frac{2 \,{\left (4 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} - 2 \,{\left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) + 6 \, x + 1}{4 \,{\left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/sqrt(x),x, algorithm="fricas")

[Out]

-1/4*(2*(4*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 8*x^2 - 2*(2*sqr
t(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)*log(2*sqrt(x)*sqrt(sqrt(x) +
1)*sqrt(sqrt(x) - 1) - 2*x + 1) + 6*x + 1)/(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sq
rt(x) - 1) - 2*x + 1)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{\sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2)/x**(1/2),x)

[Out]

Integral(sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/sqrt(x), x)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/sqrt(x),x, algorithm="giac")

[Out]

Exception raised: TypeError